In Proceedings
of the 15th International FLAIRS Conference (Special Track 'Categorization
and Concept Representation: Models and Implications'), Pensacola Florida,
May 14-17, 2002, American Association for Artificial Intelligence. (Note:
due to page limit somne things are not fully explained. For details see
forthcoming invited paper to Journal of Experimental and Theoretical
Artificial Intelligence.)

ABSTRACT: To cope with problems arising in the
description of (1) contextual interactions, and (2) the generation of new
states with new properties when quantum entities become entangled, the
mathematics of quantum mechanics was developed. Similar problems arise
with concepts. We use a generalization of standard quantum mechanics, the
mathematical lattice theoretic formalism, to develop a formal description
of the contextual manner in which concepts are evoked, used, and combined
to generate meaning.

1. The Problem of ConjunctionsAccording to the classical theory of concepts, there exists
for each concept a set of defining features that are singly necessary and
jointly sufficient (e.g. Sutcliffe 1993). Extensive evidence has
been provided against this theory (see Komatsu 1992 and Smith & Medin
1981 for overviews). Two major alternatives have been put forth. According
to the prototype theory (Rosch, 1975a, 1978, 1983; Rosch and Mervis
1975), concepts are represented by a set of, not defining, but characteristic
features, which are weighted in the definition of the prototype. A new
item is categorized as an instance of the concept if it is sufficiently
similar to this prototype. According to the exemplar theory, (e.g.,
Heit and Barsalou 1996; Medin et al. 1984; Nosofsky 1988, 1992)
a concept is represented by, not defining or characteristic features, but
a set of instances of it stored in memory. A new item is categorized
as an instance of a concept if it is sufficiently similar to one or more
of these previous instances. We use the term representational theories
to refer to both prototype and exemplar theories since concepts take the
form of fixed representations (as opposed to changing according to context).

Representational theories are adequate for predicting experimental results
for many dependent variables including typicality ratings, latency of category
decision, exemplar generation frequencies, and category naming frequencies.
However, they run into problems trying to account for the creative generation
of, and membership assessment for, conjunctions of concepts. They
cannot account for phenomena such as the so-called guppy effect,
where guppy is not rated as a good example of pet,
nor
of fish, but it is rated as a good example of
pet fish
(Osherson & Smith 1981). This is problematic because if (1) activation
of pet does not cause activation of guppy, and (2) activation
of fish does not cause activation of guppy, how is it that
(3) pet fish, which activates both pet AND
fish, causes
activation of guppy? (In fact, it has been demonstrated experimentally
that other conjunctions are better examples of the ‘guppy effect’ than
pet
fish (Storms et al. 1998), but since the guppy example is well-known
we will continue to use it here as an example.)

Zadeh (1965, 1982) tried, unsuccessfully, to solve the conjunction problem
using a minimum rule model, where the typicality of an item as a
conjunction of two concepts (conjunction typicality) equals the minimum
of the typicalities of the two constituents. Storms et al. (2000)
showed that a weighted and calibrated version of this model can account
for a substantial proportion of the variance in typicality ratings for
conjunctions exhibiting the guppy effect, suggesting the effect could be
due to the existence of contrast categories. However, another study provided
negative evidence for contrast categories (Verbeemen et al., in
press).

Conjunction cannot be described with the mathematics of classical physical
theories because it only allows one to describe a composite or joint entity
by means of the product state space of the state spaces of the two subentities.
Thus if X1 is the state space of the first subentity,
and X2 the state space of the second subentity, the state
space of the joint entity is the Cartesian product space X1 *
X2. So if the first subentity is ‘door’ and the second is
‘bell’, one can give a description of the two at once, but they are still
two. The classical approach cannot even describe the situation wherein
two entities generate a new entity that has all the properties of its subentities,
let alone a new entity with certain properties of one subentity and certain
of the properties of the other. The problem can be solved ad hoc by starting
all over again with a new state space each time there appears a state that
was not possible given the previous state space. However, in so doing we
fail to include exactly those changes of state that involve the generation
of novelty. Another possibility would be to make the state space infinitely
large to begin with. However, since we hold only a small number of items
in mind at any one time, this is not a viable solution to the problem of
describing what happens in cognition.

2. The Problem of ContextualityThe problems that arise with conjunctions reflects a more general problem
with representational theories (see Riegler, Peschl and von Stein 1999
for overview). As Rosch (1999) puts it, they do not account for the fact
that concepts "have a participatory, not an identifying function in situations";
that is, they cannot explain the contextual way in which concepts are evoked
and used (see also Gerrig and Murphy 1992; Hampton, 1987; Komatsu, 1992;
Medin and Shoben, 1988; Murphy and Medin, 1985). Not only does a concept
give meaning to a stimulus or situation, but the situation evokes meaning
in the concept, and when more than one is active they evoke meaning in
each other. It is this contextuality that makes them difficult to model
when two or more arise together, or follow one another, as in a creative
construction such as a conjunction, invention, or sentence.

3. The FormalismThis story has a precedent. The same two problems–that of conjunctions
of entities, and that of contextuality–arose in physics in the last century.
Classical physics could not describe what happens when quantum entities
interact. According to the dynamical evolution described by the Schrödinger
equation, quantum entities spontaneously enter an entangled state that
contains new properties the original entities did not have. To describe
the birth of new states and new properties it was necessary to develop
the formalism of quantum mechanics.

The shortcomings of classical mechanics were also revealed when it came
to describing the measurement process. It could describe situations where
the effect of the measurement was negligible, but not situations where
the measurement intrinsically influenced the evolution of the entity; it
could not incorporate the context generated by a measurement directly into
the formal description of the quantum entity. This too required the quantum
formalism.

First we describe the pure quantum formalism, and then we briefly describe
the generalization of it that we apply to the description of concepts.

3.1 Pure Quantum FormalismAs in any mathematical model, we begin by cutting out a piece of reality
and say this is the entity of interest, and these are its properties.
The set of actual properties constitute the state of the entity.
We also define a state space, which delineates, given how the properties
can change, the possible states of the entity. A quantum entity is described
using not just a state space but also a set of measurement contexts. The
algebraic structure of the state space is given by the vector space structure
of the complex Hilbert space: states are represented by unit vectors, and
measurement contexts by self-adjoint operators.

One says a quantum entity is entangled if it is a composite of
subentities that can only be individuated by a separating measurement.
When such a measurement is performed on an entangled quantum entity, its
state changes probabilistically, and this change of state is called quantum
collapse.

In pure quantum mechanics, if H1 is the Hilbert space
representing the state space of the first subentity, and H2
the Hilbert space representing the state space of the second subentity,
the state space of the composite is not the Cartesian product, as in classical
physics, but the tensor product, i.e., H1 Ä
H2.
The tensor product always generates new states with new properties, specifically
the entangled states. Thus it is possible to describe the spontaneous generation
of new states with new properties. However, in the pure quantum formalism,
a state can only collapse to itself with a probability equal to one; thus
it cannot describe situations of intermediate contextuality.

3.2 Generalized Quantum FormalismThe standard quantum formalism has been generalized, making it possible
to describe entities with any degree of contextuality (Aerts 1993;
Aerts and Durt 1994a, 1994b; Foulis and Randall 1981; Foulis et al.
1983; Jauch 1968; Mackey 1963; Piron 1976, 1989, 1990; Pitowsky 1989; Randall
and Foulis 1976, 1978). The generalized formalisms use lattice theory to
describe the states and properties of physical entities, and the result
is referred to as a state property system. The approach is sufficiently
general to be used to describe the different context-dependent states in
which a concept can exist, and the features of the concept manifested in
these various states.

4. Incorporating Contextuality into a Theory of ConceptsOne of the first applications of these generalized formalisms to cognition
was modeling the decision making process. Aerts and Aerts (1996) proved
that in situations where one moves from a state of indecision to a decided
state (or vice versa), the probability distribution necessary to
describe this change of state is non-Kolmogorovian, and therefore a classical
probability model cannot be used. Moreover, they proved that such situations
can
be accurately described using these generalized quantum mathematical formalisms.
Their mathematical treatment also applies to the situation where the state
of the mind changes from thinking about a concept to an instantiation of
that concept, or vice versa. Once again, context induces a nondeterministic
change of the state of the mind which introduces a non-Kolmogorivian probability
on the state space. Thus, a nonclassical (quantum) formalism is necessary.

In our approach, concepts are described using what to a first approximation
can be viewed as an entangled states of exemplars, though this is not precisely
accurate. For technical reasons (see Gabora 2001), the term potentiality
state is used instead of entangled state. For a given stimulus, the
probability that a potentiality state representing a certain concept will,
in a given context, collapse to another state representing another concept
is related to the algebraic structure of the total state space, and to
how the context is represented in this space. The state space where concepts
‘live’ is not limited a priori to only those dimensions which appear
to be most relevant; thus concepts retain in their representation the contexts
in which they have, or even could potentially be, evoked or collapsed to.
It is this that allows their contextual character to be expressed. The
stimulus situation plays the role of the measurement context by determining
which state is collapsed upon. Stimuli are categorized as instances of
a concept not according to how well they match a static prototype
or set of typical exemplars, but according to the extent to which they
correspond to, and thereby actualize or collapse upon, one the potential
interpretations of the concept. (As a metaphorical explanatory aid, if
concepts were apples, and the stimulus a knife, then the qualities of the
knife determine not just which apple to slice, but which direction to slice
through it: changing the context in which a stimulus situation is embedded
can cause a different version of the concept to be elicited.) This approach
has something in common with both prototype and exemplar theories. Like
exemplar theory, concepts consist of exemplars, but the exemplars are in
a sense ‘woven together’ like a prototype.

5. Preliminary Theoretical Evidence of the Utility
of the ApproachWe present three sources of theoretical evidence of the utility of
the approach.

5.1 A Proof that Bell Inequalities can be Violated by ConceptsThe presence of entanglement can be tested for by determining whether
correlation experiments on the joint entity violate Bell inequalities (Bell
1964). Using an example involving the concept cat and specific instances
of cats we proved that Bell inequalities are violated in the relationship
between a concept, and specific instances of this concept (Aerts et
al. 2000a; Gabora 2001). Thus we have evidence that this approach indeed
reflects the underlying structure of concepts.

5.2 Application to the Pet Fish ProblemWe have applied the contextualized approach to the Pet Fish Problem
(Aerts et al. 2000b; Gabora 2001). Conjunctions such as this are
dealt with by incorporating context-dependency, as follows: (1) activation
of pet still rarely causes activation of guppy, and likewise
(2) activation of fish still rarely causes activation of guppy.
But now (3) pet fish causes activation of the potentiality states
petin
the context ofpet fish AND fishin the context of
pet
fish. Since for both, the probability of collapsing onto the
state guppy is high, it is very likely to be activated. Thus we
have a formalism for describing concepts that is not stumped by a situation
wherein an entity that is neither a good instance of A nor B is nevertheless
a good instance of A AND B.

Note that whereas in representational approaches relations between concepts
arise through overlapping context-independent distributions, in the present
approach, the closeness of one concept to another (expressed as the probability
that its potentiality state will collapse to an actualized state of the
other) is context-dependent. Thus it is possible for two states to be far
apart from each other with respect to a one context (for example ‘fish’
and ‘guppy’ in the context of just being asked to name a fish), and close
to one another with respect to another context (for example ‘fish’ and
‘guppy’ in the context of both ‘pet’ and being asked to name a ‘fish’).
Examples such as this are evidence that the mind handles nondisjunction
(as well as negation) in a nonclassical manner (Aerts et al. 2000b).

5.3 Describing Impossibilist CreativityBoden (1990) uses the term impossibilist creativity to refer
to creative acts that not only explore the existing state space
but
transform that state space. In other words, it involves the
spontaneous generation of new states with new properties. In (Gabora 2001)
the contextual lattice approach is used to generate a mathematical description
of impossibilist creativity using as an example the invention of the torch.
This example involves the spontaneous appearance of a new state (the state
of mind that conceives of the torch) with a new property (the property
of being able to move fire).

6. Research in ProgressWe are comparing the performance of the contextualized theory of concepts
with prototype and examplar theories using previous data sets for typicality
ratings, latency of category decision, exemplar generation frequencies,
category naming frequencies on everyday natural language concepts, such
as ‘trees’, ‘furniture’, or ‘games’. The purpose of these initial investigations
is to make sure that the proposed formalism is at least as successful as
representational approaches for the simple case of single concepts. Assuming
this to be the case, we will concentrate our efforts on conjunctions of
concepts, since this is where the current approach is expected to supercede
representational theories. We will re-analyze previously collected data
for noun-noun conjunctions such as ‘pet fish’, and relative clause conjunctions
such as ‘pets that are also fish’ (Storms et al. 1996). A new study
is being prepared which will compare the proposed approach with representational
approaches at predicting the results of studies using situations that are
highly
contextual. Typicality ratings for conjunctions will be compared with,
not just their components, but with other conjunctions that share these
components. (Thus, for example, does ‘brainchild’ share features with ‘childbirth’
or ‘brainstorm’? Does ‘brainstorm’ share features with ‘birdbrain’ or ‘sandstorm’?)

REFERENCESAerts, D. 1993. Quantum Structures Due to Fluctuations of the Measurement
Situations. International Journal of Theoretical Physics 32:2207—2220.